Continuous phase transitions display, near the critical temperature, remarkable universal (i.e., independent of the specific system within a wide class of systems) macroscopic or large distance properties. Renormalization group methods allow understanding the origin of universality as well as calculating universal quantities. Universality of the large distance behaviour is then related to fixed points of the renormalization group flow. Wilson and Fisher (1972) succeeded in determining a set of fixed points
(known as Wilson-Fisher fixed points) relevant for a large class of phase transitions (liquid-vapour, Helium, ferromagnets...)
by using a method that extends to complex (i.e., non-integer) values of the space dimension \(d\) the Feynman diagram expansion, which is the standard approximation tool in perturbative quantum field theory. In this way they have defined the value of the critical exponents for non-integer values of the dimensions. They discovered that near space dimension four (\(d=4-\varepsilon\) with \(\varepsilon\rightarrow0\)), universal quantities can be calculated in the form of an \(\varepsilon\)-expansion (\(\varepsilon >0\)). This approach provided the first examples of analytic estimates of critical exponents that differed from their classical values (also known as "mean-field values" or "quasi-Gaussian values"). Quantum field theory methods then allowed for a general derivation of scaling properties and provided an efficient method for calculating universal quantities.

The renormalization group has been introduced to understand universal properties of a wide class of random or statistical systems with a large number of degrees of freedom. The term universal property is used in this context to emphasize the remarkable property that systems, which seem physically unrelated, share, somewhat unexpectedly, some non-trivial large scale properties.

The simplest examples of universality are provided, in fact, by the central limit theorem of probabilities, which deals with the average of independent random variables with the same distribution, or by the asymptotic behaviour of Markovian random walk at large times. In both problems, one is interested in the collective properties of an infinite number of random variables in a situation where the probability of large deviations with respect to the mean value decreases fast enough. In both examples, the asymptotic probability distribution is Gaussian, independently of the random variable distribution or the random walk transition function. Moreover, in the case of the random walk with finite time steps and on a space lattice, this universality allows defining a continuum limit both in time and in space. These examples do not require introducing a renormalization group because the asymptotic distributions can be calculated exactly, but a renormalization group strategy allows recovering the universal results without explicit solution.

A more intriguing situation is provided by quantum field theory.
In this context (historically started with quantum electrodynamics or QED), a naive definition of perturbation theory leads to (short-distance) infinities.
It is then necessary to modify the short-distance structure of the initial theory, leading to a finite but unphysicalregularized theory.
This modification is parametrized by some length scale known as short-distance cutoff and here denoted by \(1/\Lambda\ ,\) (\(\Lambda\) having the dimension of an inverse distance and being known as an ultraviolet (UV) cut-off because it cuts off high wavelengths).
However, and this was the surprise in the early stage of the construction of QED, a universal large distance theory, the so-called renormalized theory, can be defined by parametrizing it in terms of renormalized parameters, that are specific functions of the parameters (bare parameters) of the regularized theory. The definition of the renormalized parameters implicitly or explicitly requires the introduction of a new distance scale known as renormalization scale.
This procedure is called renormalization. The renormalized theory resulting from this process is generically no longer related to a Gaussian field distribution, which in this example corresponds to a non-interesting free field theory.
The universality of the large distance behaviour of the renormalized theory refers to the independence on the short-distance modifications of the renormalized theory when considered at scales of distance much larger than the short-distance cutoff \(1/\Lambda\) (for fixed values of the renormalized parameters).
The very existence of the renormalized theory in the "ultraviolet limit" \(\Lambda\to \infty\) (at fixed renormalized parameters) can be (perturbatively and non-perturbatively) proved for many cases of local quantum field theories, including the so-called gauge theories.

The relation between the renormalized parameters in different parametrizations of the same renormalized theory corresponding to different values of the renormalization scale was historically called renormalization group (Stueckelberg and Peterman 1953) and the linear partial differential equations for the correlators of the renormalized theory, resulting from variations of the parameters under an infinitesimal change of the renormalization scale, were called differential renormalization group equations.

Landau's theory (1937) of critical phenomena in macroscopic continuous phase transitions (and thus with divergent correlation length at the critical temperature) or mean field theory, are interpreted in modern language as corresponding to Gaussian distribution or perturbed Gaussian distribution ("quasi-Gaussian distribution"). However, for systems with short-range interactions, in space dimensions two and three, quasi-Gaussian models do not describe correctly the universal properties of phase transitions at large distance near the critical temperature. Following Kadanoff's idea, Wilson introduced a more general notion of renormalization group.
The statistical model describing the specific system is defined in terms of a microscopic scale \(1/\Lambda\ ,\) for example, the lattice spacing for a lattice model, and a Hamiltonian or configuration energy \(\mathcal{H}\ .\) The idea is to recursively integrate out the short distance degrees of freedom of the system to generate a sequence of effective Hamiltonians \(\mathcal{H}_{\lambda}\) corresponding to increasing scales \(1/(\lambda\Lambda)\ ,\) with \(\lambda\in[0,1]\ .\) The (non-linear) flow equations that relate Hamiltonians associated to different scales are also called renormalization group equations. In a continuum space (but not on the lattice), infinitesimal changes of scales are possible and lead to differential equations for the effective Hamiltonian, see (1). If the renormalization group has attractive fixed points, then universality of the large distance properties can be understood since the effective Hamiltonians eventually converge as \(\lambda\to 0 \) toward a fixed-point Hamiltonian. The basin of attraction of a given fixed point in the space of Hamiltonians is called a universality class.
It can be verified that the Gaussian theory provides the simplest example of a fixed point, thus called Gaussian fixed point.

It is now understood that both the field theory renormalization group and Wilson's general formulation are related. The field renormalization group is the asymptotic form of the general renormalization group in a neighbourhood of the Gaussian fixed point.

Statistical field theory

Even if the initial statistical model is defined in terms of random variables associated to the sites of a space lattice, and taking only a finite set of values (like, e.g., the classical spins of the Ising model), it is intuitive that the fixed point theory will be a statistical field theory in continuum space. Therefore, we consider a classical statistical system defined in terms of a random real field \(\phi(x)\) in continuum space, \(x\in\mathbb{R}^d\ .\) The partition function is then given by a field integral (i.e., a sum over field configurations)
\[\mathcal{Z}=\int[\mathrm{d}\phi(x)]\,\mathrm{e}^{-\mathcal{H}(\phi)}.\]
\(\mathcal{H}(\phi)\) is called the Hamiltonian in classical statistical physics language. The condition of short range interactions translates into the property of locality of the field theory\[\mathcal{H}(\phi)\] can be chosen as a space-integral over a linear combination of monomials in the field and its derivatives.
We assume also space translation and rotation invariance, and \(\mathbb{Z}_2\) reflection symmetry\[\mathcal{H}(\phi) = \mathcal{H}(-\phi)\] except when stated explicitly otherwise.
Finally, the coefficients of \(\mathcal{H}(\phi)\) are regular functions of the temperature \(T\) near the critical temperature \(T_c\) where a continuous phase transition occurs. In the low temperature phase, the \(\mathbb{Z}_2\) symmetry is spontaneously broken.

Physical observables involve field correlation functions (generalized moments),
\[\langle \phi(x_1)\phi(x_2)\ldots\phi(x_n)\rangle\equiv {1\over\mathcal{Z}}\int[\mathrm{d}\phi(x)]\phi(x_1)\phi(x_2)\ldots\phi(x_n)\,\mathrm{e}^{-\mathcal{H}(\phi)}.\]
They can be derived by functional differentiation from a generalized partition function in an external field \(H(x)\ ,\)
\[\mathcal{Z}(H)=\int[\mathrm{d}\phi(x)]\,\exp\left[-\mathcal{H}(\phi)+\int\mathrm{d}^d x\,H(x)\phi(x)\right],\]
as
\[\langle \phi(x_1)\phi(x_2)\ldots\phi(x_n)\rangle={1\over \mathcal{Z}(0)}\left.{\delta\over\delta H(x_1)}{\delta\over\delta H(x_2)}\ldots {\delta\over\delta H(x_n)}\mathcal{Z}(H)\right|_{H=0}\,.\]
The more direct physical observables are the connected correlation functions \( W^{(n)}(x_1,x_2,\ldots,x_n)\) (generalized cumulants), which can be obtained by function differentiation from the free energy \(\mathcal{W}(H)=\ln\mathcal{Z}(H)\) (omitting a temperature factor irrelevant here) in the external field \(H\ :\)
\[W^{(n)}(x_1,x_2,\ldots,x_n)= \left.{\delta\over\delta H(x_1)}{\delta\over\delta H(x_2)}\ldots {\delta\over\delta H(x_n)}\mathcal{W}(H)\right|_{H=0}\,.\]
Due to translation invariance, \(W^{(n)}(x_1,x_2,\ldots,x_n)=W^{(n)}(x_1+a,x_2+a,\ldots,x_n+a)\) for any vector \(a\ .\)

Connected correlation functions have the so-called cluster property: if one separates the points \(x_1,\ldots,x_n\) in two non-empty sets, connected functions go to zero when the distance between the two sets goes to infinity. It is the large distance behaviour of connected correlation functions in the critical domain near \(T_c\) that may exhibit universal properties.

The renormalization group: General formulation

The denomination renormalization group (RG) refers to the property that \(\ln \lambda\ ,\) where \(\lambda>0\) belongs to the dilatation (associated to the change of scale) semi-group, belongs to the additive group of real numbers.

To construct an RG flow, the basic idea is to integrate in the field integral recursively over short distance degrees of freedom. This leads to the definition of an effective Hamiltonian \(\mathcal{H}_{\lambda}\ ,\) function of a scale parameter \(\lambda\) (such that \( \mathcal{H}_1=\mathcal{H}\)) and of a transformation \(\mathcal{T}\) in the space of Hamiltonians such that
\[\tag{1}
\lambda{ \mathrm{d} \over \mathrm{d} \lambda}\mathcal{H}_{\lambda} = \mathcal{T} \left[\mathcal{H}_{\lambda}\right],\]

an equation called RG equation (RGE).
The appearance of the derivative \(\lambda \mathrm{d}/\mathrm{d}\lambda=\mathrm{d}/\mathrm{d}\ln\lambda\) reflects the multiplicative character of dilatations or scale changes. The RGE thus defines a dynamical process in the "time" \(\ln \lambda\ .\)

We assume that the mapping \( \mathcal{H}_1\mapsto \mathcal{H}_{\lambda}\) is Markovian, that is, that \(\mathcal{T} \left[\mathcal{H}_{\lambda}\right]\) depends on \(\mathcal{H}_{\lambda}\) but not on the trajectory that has led from \(\mathcal{H}_{\lambda=1}\) to \(\mathcal{H}_{\lambda},\) that the Markovian process is stationary, in such a way that \(\mathcal{T} \left[\mathcal{H}_{\lambda}\right]\) depends on \(\lambda\) only through
\(\mathcal{H}_{\lambda}\) (and thus does not depend on \(\lambda\) explicitly). Finally, we assume, and this is also an important hypothesis, that the mapping \(\mathcal{T}\) is sufficiently differentiable (for example, infinitely differentiable). These assumptions summarize the characteristic properties of these equations and are at the origin of many renormalization group properties.

Universality is related to the existence of fixed points solution of the equation
\[\mathcal{T}(\mathcal{H}^*)=0\,.\]
Since the mapping \(\mathcal{T}\) is differentiable, the local flow near a fixed point can then be studied by linearizing equation (1) at the fixed point:
\[\mathcal{T}(\mathcal{H}^*+\Delta\mathcal{H}_\lambda)\sim L^*\Delta\mathcal{H}_\lambda\,,\]
and is governed by the eigenvalues and eigenvectors of the linear operator \( L^*\ .\) Formally, the solution of the linearized equations can be written as
\[\mathcal{H}_\lambda =\mathcal{H}^*+\lambda^{L^*}\left(\mathcal{H}_{\lambda=1}-\mathcal{H}^*\right).\]

The Gaussian fixed point

An RG flow can be constructed that has as a fixed point the critical
Gaussian model corresponding, in \(d \) space dimensions, to the quadratic Hamiltonian
\[\tag{2}
\mathcal{H}^*_{\mathrm{G}}(\phi)={1\over2}\int\mathrm{d}^d x\sum_{\mu=1}^d\bigl(\partial_\mu\phi(x)\bigr)^2\]

(\(\partial_\mu\equiv \partial/\partial x_\mu\)) and, thus, to a free massless field theory in quantum field theory language.

The Hamiltonian flow near the Gaussian fixed point, in the linear approximation, can be implemented by the simple scaling
\[\tag{3}
\phi(x)\mapsto \lambda^{(2-d)/2}\phi( x/\lambda).\]

After the change of variables \(x'=x/\lambda\ ,\) one verifies that \(\mathcal{H}^*_{\mathrm{G}}(\phi)\) is indeed invariant.

For Ising-like systems with a \(\mathbb{Z}_2\) symmetry or, more generally, for models with an \(O(N)\) symmetry, it is found that the Gaussian fixed point is stable above space dimension four, marginally stable in dimension four and unstable below dimension four.

Short distance regularization

However, with the Hamiltonian (2), the Gaussian model (6) has a problem: too singular fields contribute to the corresponding field integral in such a way that correlation functions at coinciding points are not defined. For example,
\[W^{(2)}(0,0)={1\over(2\pi)^d}\int{\mathrm{d}^d p\,\ \over p^2}\,,\]
which diverges in any space dimension \(d \ge2\ .\) In particular, expectation values of local perturbations to the Gaussian theory of the form \(\phi^m(x)\) are not defined.
Therefore, it is necessary to modify the Gaussian model at short distance to restrict the field integration to more regular fields, continuous to define powers of the field, satisfying differentiability conditions to define expectation values of the field and its derivatives taken at the same point. This can be achieved by adding terms with enough more derivatives
\[\tag{4}
\mathcal{H}^*_{\mathrm{G}}(\phi)\mapsto \mathcal{H}_{\mathrm{G}}(\phi)=\mathcal{H}^*_{\mathrm{G}}(\phi)+{1\over2}\sum_{k=2} \alpha_k\int\mathrm{d}^d x\,\phi(x)\nabla_x^{2k} \phi(x),\]

where the coefficient \(\alpha_k\) are only constrained by the positivity of the Hamiltonian. One verifies by explicit calculation that, for \(d>2\) the modified Gaussian theory has the same large distance behaviour but is now regularized at short distance. For \(d\le 2\ ,\) correlation functions still do not exist due now to a low momentum, large distance singularity, showing that critical phenomena can then certainly not be described by the Gaussian model.

The linearized RG flow

The transformation (3) generates the linearized RG flow at the Gaussian fixed point.
Eigenvectors of the linear flow (3) are monomials of the form \(\mathcal{O}_{n,k}(\phi)=\int\mathrm{d}^d x\,O_{n,k}(\phi,x) \ ,\) where \(O_{n,k}(\phi,x)\) is a product of powers of the field and its derivatives at point \(x\) with \(2n\) powers of the field (reflection \(\mathbb{Z}_2\) symmetry) and \(2k\) powers of \(\partial_\mu\ .\) Their RG behaviour under the transformation (3) is then given by a simple dimensional analysis.
One defines the dimension of \(x\) as -1. The dimension of \(\partial_\mu\) is then +1. The Gaussian dimension of the field is \([\phi]=(d-2)/2\ .\) The dimension \([\mathcal{O}_{n,k}]\) of \(\mathcal{O}_{n,k}\) is then
\[\tag{5}
[\mathcal{O}_{n,k}] =-d+n(d-2)+2k\,.\]

It can be verified that \(\mathcal{O}_{n,k}\) scales like \(\lambda^{-[\mathcal{O}_{n,k}]},\) and the corresponding eigenvalue of \(L^*\) thus is \(\ell_{n,k}=-[\mathcal{O}_{n,k}]\ .\) When \(\lambda\to+\infty,\) for \(\ell_{n,k}>0\) the amplitude of \(\mathcal{O}_{n,k}(\phi)\) increases; it is a direction of instability and in the RG terminology \(\mathcal{O}_{n,k}(\phi)\) is a relevant perturbation. For \(\ell_{n,k}<0\ ,\) the amplitude of \(\mathcal{O}_{n,k}(\phi)\) decreases; it is a direction of stability and in the RG terminology \(\mathcal{O}_{n,k}(\phi)\) is an irrelevant perturbation. In the special case \(\ell_{n,k}=0\ ,\) one speaks of a marginal perturbation and the linear approximation is no longer sufficient to discuss stability. Logarithmic behaviour in \(\lambda\) is then expected.

One verifies that \(\int\mathrm{d}^d x\,\phi^2(x)\) corresponds always to a direction of instability: it induces a deviation from the critical temperature and thus a finite correlation length (or a non-vanishing mass in field theory language). For \(d>4\ ,\) no other perturbation is relevant and the Gaussian fixed point is stable on the critical surface. At \(d=4\ ,\) one term becomes marginal\[\int\mathrm{d}^d x\,\phi^4(x)\ ,\] which below dimension four becomes relevant. In dimension \(d=4-\varepsilon\ ,\) \(\varepsilon\) positive and small (a notion we define later), it is the only relevant perturbation and one expects to be able to describe critical properties with a Gaussian theory to which this unique term is added.

If what follows, we assume that initially the statistical system is very close to the Gaussian fixed point. The RG flow is then first governed by the linear flow. Therefore, we implement first the corresponding RG transformation. We introduce a parameter \(\Lambda\gg 1\) and substitute \(\phi(x)\mapsto \Lambda^{(2-d)/2}\phi( x/\Lambda).\)
After the change of variables \(x'=x/\Lambda\ ,\) the monomials \(\mathcal{O}_{n,k}(\phi)\) are multiplied by \(\Lambda^{-[\mathcal{O}_{n,k}]}\ .\) In the quantum field theory language, this could be called a Gaussian renormalization. The introduction of \(\Lambda\) has the effect of expressing the dimension (5) in terms of \(\Lambda\ :\) space coordinates \(x\) have dimension \(\Lambda^{-1},\) derivatives dimension \(\Lambda\) and the field dimension \(\Lambda^{(d-2)/2}.\) The Hamiltonian is dimensionless.

The Gaussian model

The Gaussian model is obtained by adding to the Gaussian Hamiltonian (4) the only relevant term above dimension four:
\[\tag{6}
\mathcal{H}_0(\phi)=\mathcal{H}_{\mathrm{G}}(\phi)+ {1\over2}\alpha_0 \Lambda^2\int\mathrm{d}^d x\,\phi^2(x),\]

where \(\alpha_0 \) is the amplitude of the relevant term and is non-negative in the Gaussian model. The Gaussian model can only describe the high temperature phase \(T\ge T_c\ .\)
In a Gaussian model, all correlation functions can be expressed in terms of the two-point function with the help of Wick's theorem. Except at coinciding points, one can take the \(\Lambda\to\infty\) limit. However, to obtain a non-trivial universal large distance behaviour, it is also necessary to compensate the RG flow by choosing \(\alpha_0\) infinitesimal, taking the \(\Lambda\to\infty\) limit at \( r =\alpha_0 \Lambda^2\) fixed (\(r\) is a renormalized parameter in quantum field theory language). This defines the critical domain. The two-point function in the critical domain can then be written as
\[W^{(2)}(x_1,x_2)={1\over(2\pi)^d}\int{\mathrm{d}^d p\,\mathrm{e}^{i p(x_1-x_2)} \over p^2+r}\,.\]
It has the so-called Ornstein-Zernicke form, also found in mean-field theory. Comparing with mean-field theory, one concludes that \(r\) plays the role of the deviation from the critical temperature \(T_c\) near \(T_c\ :\) \(r\propto T-T_c\) for \(T-T_c\to 0\ .\)

For \(r>0\ ,\) \(W^{(2)}(x,0)\) decays exponentially at large distance as
\[W^{(2)}(x,0)\mathop{\propto}_{|x|\to\infty}{\mathrm{e}^{-|x|/\xi}\over |x|^{(d-1)/2}},\]
where the parameter \(\xi\) that governs the decay rate is called the correlation length. Here, \(\xi=1/\sqrt{r}\propto(T-T_c)^{-\nu}\) with \(\nu=1/2\ ,\) where \(\nu\) is the correlation length exponent.
At \(r=0\) (i.e., \(T=T_c\)) and for \(d>2\ ,\) \(W^{(2)}(x,0)\) decays algebraically as
\[W^{(2)}(x,0)\mathop{\propto}_{|x|\to\infty}|x|^{2-d}.\]
In two dimensions, the Gaussian model is not defined at \(T_c\ .\)

The perturbed Gaussian or quasi-Gaussian model

To allow for spontaneous symmetry breaking and, thus, to be able to describe physics below \(T_c\ ,\) terms have necessarily to be added to the Gaussian Hamiltonian (4) to generate a double-well potential for constant fields. The minimal addition, which is also the leading term from the RG viewpoint, is of \(\phi^4\) type. This leads to
\[\mathcal{H}(\phi)= \mathcal{H}_{\mathrm{G}}(\phi)+{g\over4!}\Lambda^{4-d}\int\mathrm{d}^d x\,\phi^4(x), \quad g>0\,.\]
The \(\phi^4\) term generates a shift of the critical temperature. To recover a critical theory (\(T=T_c\)), it is necessary to further add a \(\phi^2\) term with a specific \(g\)-dependent coefficient:
\[\mathcal{H}(\phi) \mapsto \mathcal{H}(\phi)+{1\over2}(\alpha_0)_c(g)\Lambda^2\int \mathrm{d}^d x\,\phi^2(x), \]
a mass renormalization in quantum field theory terminology.
For dimensions \(d>4\ ,\) the \(\phi^4\) term is an irrelevant contribution that does not invalidate the universal predictions of the Gaussian model and corrections to the Gaussian theory can be obtained by expanding in powers of the coefficient \(g\ .\) For example, setting \(u=g\Lambda^{4-d},\) the partition function is then given by
\[\mathcal{Z}=\sum_{k=0}^\infty{(-u)^k \over (4!)^k k!}\left\langle\left(\int\mathrm{d}^d x\,\phi^4(x)\right)^k\right\rangle_{\mathrm{G}}.\]
The Gaussian expectations values \(\langle\bullet\rangle_{\mathrm{G}}\) can then be evaluated in terms of the Gaussian two-point function with the help of Wick's theorem.

By contrast, for \(d<4 \ ,\) the \(\phi^4\) contribution is relevant: the Gaussian fixed point is unstable and no longer governs the large distance behaviour. The perturbative expansion of the critical theory (\(T=T_c\)) in powers of \(u\) contains so-called infra-red, that is, long distance, or small momentum in Fourier space, divergences.
To determine the large distance behaviour of correlation functions, it becomes necessary to construct a general renormalization group. This leads to functional equations that, in general, cannot be solved analytically. However, a trick has been discovered to extend the definition of all terms of the perturbative expansion to arbitrary complex values of the dimension \(d\) in the form of meromorphic functions. This allows replacing, in dimension \(d=4-\varepsilon\ ,\) the general renormalization group by a much simpler asymptotic form and studying the model analytically as an expansion in powers of \(\varepsilon\ .\)

where, in analogy with quantum mechanics, the Fourier variables \( p_i\) are called momenta (and have dimension \(\Lambda\)).
We also introduce the Fourier representation of the Gaussian two-point function (or propagator) \(\Delta(x)\ ,\) corresponding to the Hamiltonian (4),
\[\Delta(x)\equiv \langle \phi(x)\phi(0)\rangle_{\mathrm{G}}={1\over(2\pi)^d}\int\mathrm{d}^d p\,\mathrm{e}^{-ipx}\tilde \Delta(p).\]

Dimensional continuation

A general representation of the Gaussian two-point function useful for dimensional continuation is
\[\tag{8}
\tilde \Delta(p)=\int_0^\infty\mathrm{d}s\,\rho(s\Lambda^2)\mathrm{e}^{-s p^2}, \]

where \(\rho(s)\to 1\) when \(s\to \infty\ .\) To reduce the field integration to continuous fields and, thus, to render the perturbative expansion finite, one needs at least \(\rho(s)=O(s^q)\) with \(q>(d-2)/2\) for \(s\to0\ .\) If one wants the expectation values of all local polynomials to be defined, one must impose to \(\rho(s)\) to converge to zero faster than any power. In the context of quantum field theory, since the effect of the \(\rho\)-factor is to suppress \(\tilde \Delta(p)\) for values of \(|p|\gg\Lambda\ ,\) \(\Lambda\) is called the cut-off. A contribution to perturbation theory (represented graphically by a Feynman diagram) takes, in Fourier representation, the form of a product of propagators integrated over a subset of momenta. With the representation (8), all momentum integrations become Gaussian and can be performed, resulting in explicit analytic meromorphic functions of the dimension parameter \(d\ .\) For example, the contribution of order \(g\) to the two-point function is proportional to
\[\Omega_d={1\over(2\pi)^d}\int\mathrm{d} p\,\tilde \Delta(p)={1\over(2\pi)^d}\int\mathrm{d} p\,\int_0^\infty\mathrm{d}s\,\rho(s\Lambda^2)\mathrm{e}^{-s p^2}={1\over(4\pi)^{d/2}}\int_0^\infty\mathrm{d}s\,s^{-d/2}\rho(s\Lambda^2) ,\]
which in the latter form is holomorphic for \(2<\mathrm{Re}\, d<2(1+q).\)

Dimensional regularization

While for the theory of critical phenomena, dimensional continuation is sufficient since it allows exploring the neighbourhood of dimension four, for practical calculations restricted to leading order at large distance, an additional step is useful. It can be verified that if one takes \(\mathrm{Re}\, d\) sufficiently small so that by naive power counting all momentum integrals are convergent, one can then, after explicit dimensional continuation, take the infinite \(\Lambda\) limit. The resulting perturbative contributions become meromorphic functions with poles, in particular at dimensions at which large momentum, and low momentum in the critical theory, divergences appear.
This method of regularizing large momentum divergences is called dimensional regularization and is extensively used in quantum field theory. It has also been used to calculate universal quantities in the theory of critical phenomena, like critical exponents, as \(\varepsilon=4-d\)-expansions.

Perturbative renormalization group

The renormalization theorem

The perturbative renormalization group, as it has been developed in the framework of the perturbative expansion of quantum field theory, relies on the so-called renormalization theory. For the \(\phi^4\) field theory it has been first formulated in space dimension \(d=4\ .\) For critical phenomena, a small extension is required that involves an additional expansion in powers of \(\varepsilon=4-d\ ,\) after dimensional continuation.

To formulate the renormalization theorem, one introduces
a momentum \(\mu\ ,\) called the renormalization scale, and a parameter \(g_{\mathrm{r}} \) characterizing the effective \(\phi^4\) coefficient at scale \(\mu\ ,\) called the renormalized coupling constant.
One can then find two dimensionless functions \(Z(\Lambda /\mu,g)\) and \(Z_g(\Lambda /\mu,g),\) that satisfy (\(g\) and \(\Lambda/\mu\) are the only two dimensionless combinations)
\[\tag{9}
\Lambda ^{4-d}g= \mu^{4-d}Z_g(\Lambda /\mu,g)g_{\mathrm{r}}=\mu^{4-d} g_{\mathrm{r}} +O(g^2),\ Z(\Lambda /\mu,g)=1+O(g),\]

calculable order by order in a double series expansion in powers of \(g\) and \(\varepsilon\ ,\) such that all connected correlations functions
\[\tag{10}
\tilde W^{(n)}_{\mathrm{r}} (p_i ;g_{\mathrm{r}} ,\mu ,\Lambda )=Z
^{-n/2} (g,\Lambda / \mu )\tilde W^{(n)} (p_i ;g,\Lambda ), \]

called renormalized,
have, order by order in \(g_{\mathrm{r}}\ ,\) finite limits \(\tilde W^{(n)} _{\mathrm{r}} (p_i ;g_{\mathrm{r}},\mu)\) when \(\Lambda \to\infty\) at \(p_i,\mu,g_{\mathrm{r}}\) fixed. The factor \(Z^{1/2}(\Lambda /\mu,g)\) is a multiplicative correction to the Gaussian field rescaling factor \(\Lambda^{(d-2)/2}.\)

Remarks

There is some arbitrariness in the choice of the renormalization constants \(Z\) and \(Z_g\) since they can be multiplied by arbitrary functions of \(g_{\mathrm{r}}.\) The constants can be completely determined by imposing three renormalization conditions to the renormalized correlation functions, which are then independent of the specific choice of the regularization. This a first important result: since initial and renormalized correlation functions have the same large distance behaviour, this behaviour is to a large extent universal since it can, therefore, only depend at most on one parameter, the \(\phi^4\) coefficient \(g\ .\)

The renormalization constant \(Z^{1/2}\) is just the ratio between the Gaussian field renormalization and the renormalization in presence of the \(\phi^4\) interaction.

From equation (9), one immediately infers that \(\beta(g)=-\varepsilon g+O(g^2).\)

RG equations in the critical domain above \(T_c\)

Correlation functions may also exhibit universal properties near \(T_c\) when the correlation length \(\xi\) is large in the microscopic scale, here, \(\xi\Lambda\gg 1\ .\)
To describe universal properties in the critical domain above \(T_c\ ,\) one adds the \(\phi^2\) relevant term to the Hamiltonian:
\[\mathcal{H}_t(\phi)=\mathcal{H}(\phi)+{t\over2} \int\mathrm{d}^d x\,\phi^2(x),\]
where \(t\ ,\) the coefficient of \(\phi^2\ ,\) characterizes the deviation from the critical temperature\[t\propto T-T_{\rm c}.\]
The renormalization theorem leads to the appearance of a new renormalization factor \(Z_2(\Lambda/\mu, g)\) associated with the parameter \(t.\) By arguments of the same nature as in the critical theory, one derives a more general RG equation of the form (Zinn-Justin 1973)
\[\tag{13}
\left[ \Lambda{ \partial \over \partial \Lambda} +\beta(g){\partial \over
\partial g}+{n \over 2}\eta(g)-\eta_2(g)t{\partial \over \partial t} \right]
\tilde W^{(n)}_{\mathrm{as.}} (p_i ;t,g,\Lambda )=0\,,\]

where a new RG function \(\eta_2(g)\) related to \(Z_2(\Lambda/\mu, g)\) appears.

These equations can be further generalized to deal with an external field (a magnetic field for magnetic systems) and the corresponding induced field expectation value (magnetization for magnetic systems). An RG equation for the equation of state follows.

Renormalized RG equations

For \(d<4\ ,\) if one is only interested in the leading scaling behaviour (and the first correction), it is technically simpler to use dimensional regularization and the renormalized theory in the so-called minimal (or modified minimal) subtraction scheme. Equation (10) is asymptotically symmetric between initial and renormalized correlations. One thus derives also (for the critical theory)
\[\left[ \mu{ \partial \over \partial \mu} +\tilde \beta (g_{\mathrm{r}}
){\partial \over \partial g_{\mathrm{r}}}+{n \over 2}\tilde \eta (g
_{\mathrm{r}} ) \right]\tilde W^{(n)}_{\mathrm{r}}(p_i,g_{\mathrm{r}} ,\mu)=0 \]
with the definitions
\[\tilde \beta (g_{\mathrm{r}} ) = \left.\mu{ \partial
\over \partial\mu} \right|_g g_{\mathrm{r}}\,, \quad \tilde \eta
(g_{\mathrm{r}} ) = \left.\mu{ \partial \over \partial \mu} \right|_g \ln
Z (g_{\mathrm{r}},\varepsilon)\,.\]
In this scheme, the renormalization constants (9) are obtained by going to low dimensions where the infinite \(\Lambda\) limit, at \(g_{\mathrm{r}}\) fixed, can be taken. For example,
\[\lim_{\Lambda\to\infty}\left.Z(\Lambda/\mu,g)\right|_{g_{\mathrm{r}}\ \mathrm{fixed}}=Z (g_{\mathrm{r}},\varepsilon).\]
Then, order by order in powers of \(g_{\mathrm{r}}\ ,\) they have a Laurent expansion in powers of \(\varepsilon\ .\) In the minimal subtraction scheme, the freedom in the choice of the renormalization constants is used to reduce the Laurent expansion to the singular terms. For example, \(Z (g_{\mathrm{r}},\varepsilon) \) takes the form
\[Z (g_{\mathrm{r}},\varepsilon) =1+\sum_{n=1}^\infty {\sigma_n(g_{\mathrm{r}}) \over \varepsilon^n}\]
with \(\sigma_n(g_{\mathrm{r}}) =O(g_{\mathrm{r}}^{n+1}).\)
A remarkable consequence is that the RG functions \(\tilde \eta(g_{\mathrm{r}} )\ ,\) and \(\tilde \eta_2(g_{\mathrm{r}} )\) when a \(\phi^2\) term is added, become independent of \(\varepsilon\) and \(\tilde \beta (g_{\mathrm{r}} )\) has the simple dependence
\[\tilde\beta (g_{\mathrm{r}} )=-\varepsilon g_{\mathrm{r}} +\tilde\beta_2(g_{\mathrm{r}} ), \]
where \(\tilde\beta_2(g_{\mathrm{r}} )=O(g_{\mathrm{r}}^2) \) is also independent of \(\varepsilon\ .\)

The exponent \(d_\phi=(d-2+\eta)/2\) is the dimension of the field \(\phi\ ,\) from the point of view of large distance properties.

Wilson's-Fisher fixed point: Epsilon-expansion

The Ising class fixed point

For practical RG calculations, it is more convenient to deal with vertex functions than connected correlation functions. The generating functional of vertex functions is obtained from the generating functional of connected functions by a Legendre transformation that generalizes the relation between free energy and thermodynamic potential. From the viewpoint of Feynman diagrams, vertex functions are one-line irreducible.
For the two-point and four-point functions relevant here, the vertex functions \(\tilde \Gamma^{(n)}\)
in Fourier representation above \(T_c\) are given by
\[\tilde\Gamma^{(2)}(p)=1/\tilde W^{(2)}(p),\quad \tilde\Gamma^{(4)}(p_1,p_2,p_3,p_4)=\left.\tilde W^{(4)}(p_1,p_2,p_3,p_4)\right/\prod_{i=1}^4 \tilde W^{(2)}(p_i).\]
The RG equations satisfied by the asymptotic vertex functions are obtained from equations (12) or (13) by simply changing \(\eta(g)\) into \(-\eta(g).\)

At order \(g\ ,\) in the critical theory one finds that \(\tilde\Gamma^{(2)}\) is not modified, the order \(g\) contribution (first diagram of Figure 1) being a constant that only renormalizes the critical value \(r_c=(\alpha_0)_c(g)\Lambda^2\ :\)
\[\tilde\Gamma^{(2)}(p,g,\Lambda)=p^2+O(g^2).\]
Applying the RG equation, one obtains \(\eta(g)=O(g^2).\)
The four-point vertex function at order \(g^2\) is then given by
\[\tilde\Gamma^{(4)}(p_1,p_2,p_3,p_4,g,\Lambda)=\Lambda^{\varepsilon} g-\textstyle{{1\over2} }g^2\left[B_d(p_1+p_2)+B_d(p_1+p_3)+B_d(p_1+p_4)\right]+O(g^3,g^2\varepsilon),\]
where (second diagram of Figure 1)
\[B_d(p)={1\over(2\pi)^d}\int\mathrm{d}^d q\,\tilde\Delta(q)\tilde\Delta(p-q).\]
In the logic of the \(\varepsilon\)-expansion, at leading order one needs only \(B_4\ .\) Without the cut-off \(\Lambda\ ,\) the integral would then diverge logarithmically. For \(\Lambda\) large, it is thus dominated by
\[B_4(p)\sim{1\over(2\pi)^d}\int_{1<|q|<\Lambda}{\mathrm{d}^4 q \over q^4}\sim N_4 \ln\Lambda\,\]
where \(N_d\) is the loop factor,
\[\tag{15}
N_d= {2\over(4\pi)^{d/2}\Gamma(d/2)},\]

which, at higher orders, it is convenient not to expand in powers of \(\varepsilon\ .\)
Then,
\[\tilde\Gamma^{(4)}(p_1,p_2,p_3,p_4,g,\Lambda)= g+g\varepsilon \ln\Lambda-\textstyle{{3\over2} }g^2\left(\ln\Lambda+\ \mathrm{finite}\right)+O(g^3,g^2\varepsilon,g\varepsilon^2).\]
\(\tilde\Gamma^{(4)}\) satisfies an equation that can inferred from equation (12),
\[\left[ \Lambda{ \partial \over \partial \Lambda} +\beta
(g ){\partial \over \partial g}-2\eta (g) \right] \tilde \Gamma_{\mathrm {as.}}^{(4)}
(p_i ;g,\Lambda )=0\,.\]
Applying it to the explicit expansion, one concludes
\[\beta(g)=-\varepsilon g+{3\over 16\pi^2} g^2+O(g^3,\varepsilon g^2).\]
In the sense of an \(\varepsilon\)-expansion, \(\beta(g)\) thus has a zero \(g^*\) with a positive slope (Wilson-Fisher's fixed point 1972),
\[ g^*=16\pi^2\varepsilon/3+O(\varepsilon^2),\quad\omega=\beta'(g^*)=\varepsilon+O(\varepsilon^2),\]
which governs the large momentum behaviour of correlation functions. In addition, the exponent \(\omega\) governs the leading correction to the critical behaviour.

Generalization

The results obtained for models with a \(\mathbb{Z}_2\) reflection symmetry can easily be generalized to \(N\)-vector models with \(O(N)\) group (rotations-reflections in \(N\) space dimensions) symmetry, which belong to different universality classes. Their universal properties can then be derived from an \(O(N)\) symmetric field theory with an \(N\)-component field \(\boldsymbol{\phi}(x)\) and a \(g(\boldsymbol{\phi}^2)^2\) quartic term. Further generalizations involve theories with \(N\)-component fields but smaller symmetry groups, such that several independent quartic \(\phi^4\) terms are allowed. The structure of fixed points may then be more complicate.

Epsilon expansion: A few results

From the simple existence of the fixed point and of the corresponding \(\varepsilon\)-expansion, universal properties for a large class of critical phenomena can be proved to all orders in \(\varepsilon\ :\) this includes scaling relations between critical exponents, scaling behaviour of correlation functions or the equation of state... Moreover, universal quantities can be calculated as \(\varepsilon\)-expansions.

General results

An example of the general results that can be obtained is provided by the equation of state of magnetic systems, that is, the relation between applied magnetic field \(H\ ,\) magnetization \(M\) and temperature \(T\ .\) In the relevant limit \(|H|\ll 1\ ,\) \(|T-T_c\ll 1\ ,\) RG allows proving Widom's conjectured scaling form
\[H=M^\delta f\bigl((T-T_c)/M^{1/\beta}\bigr),\]
where \(f(z)\) is a universal (up to normalizations) calculable (in an \(\varepsilon\)-expansion) function. Moreover, the exponents satisfy the relations
\[\delta={d+2-\eta\over d-2+\eta},\quad \beta =\textstyle{{1\over2} }\nu(d-2+\eta),\]
where \(\nu\ ,\) the correlation length exponent, given by
\[\nu=1/\bigl(\eta_2(g^*)+2\bigr),\]
characterizes the divergence \(\xi\) of the correlation length at \(T_c\ :\) \(\xi\propto|T-T_c|^{-\nu}.\)

Other relations can be derived, involving the magnetic susceptibility exponent \(\gamma\) characterizing the divergence of the two-point correlation function at zero momentum at \(T_c\) or the behaviour of the specific heat \(\alpha\ :\)
\[\gamma=\nu(2-\eta),\ \alpha =2 -\nu d\,.\]
Note the relations involving the dimension \(d\) explicitly are not valid for the Gaussian fixed point.

Critical exponents as epsilon-expansions

We give here the results for all \(N\)-vector models, the results for the \(\mathbb{Z}_2\) models being recovered by setting \(N=1\ .\) Although the RG functions of the \((\boldsymbol{\phi}^2 )^2\) field theory are known to five-loop order and, thus, the critical exponents are known up to \(\varepsilon^5\ ,\) for illustration purpose we give here only two successive terms in the expansion, referring
to the literature for higher order results. In terms of the variable \(v =N_d \, g\) where \(N_d\) is the loop factor
(15), the RG functions \(\beta(v)\) and \(\eta_2(v)\) at two-loop order, \(\eta(v)\)
at three-loop order are
\[\beta(v) = -\varepsilon v+ {(N+8) \over 6}v^2 -{(3N+14) \over
12} v^3 +O(v^4),\ :\]

Though this may not be obvious on these few terms, the \(\varepsilon\)-expansion is divergent for any \(\varepsilon>0\ ,\) as large order calculations based on instantons have shown. Extracting precise numbers from the known terms of the series requires a summation method.
For example, adding simply the known successive terms for \(\varepsilon=1\) and \(N=1\) yields
\[ \gamma =1.000\ldots\,,\ 1.1666\ldots\,,\ 1.2438\ldots\,,1.1948\ldots\,,\ 1.3384\ldots\,,\ 0.8918\ldots\,,\]
while the best field theory estimate is \(\gamma=1.2396\pm0.0013\ .\)

Summation of the epsilon-expansion and numerical values of exponents

We display below (Table Figure 2) the results for some critical exponents of the \(O(N)\) model obtained from Borel summation of the \(\varepsilon\)-expansion (Guida and Zinn-Justin 1998). Due to scaling relations like \(\gamma=\nu(2-\eta),\) \(\gamma+2\beta=\nu d\ ,\) only two among the first four are independent, but the series are summed independently to check consistency. \(N=0\) corresponds to statistical properties of polymers (mathematically the self-avoiding random walk), \(N=1\) to the Ising universality class, to liquid-vapour, binary mixtures or anisotropic magnet phase transitions. \(N=2\) describes the superfluid Helium transition, while \(N=3\) correspond to isotropic ferromagnets.

However, perturbative calculations directly in three dimensions, based on the RG formalism of Callan-Symanzik equations, as suggested initially by Parisi, have been extended to higher orders than the \(\varepsilon\)-expansion by Nickel and provided more precise results for critical exponents and the equation of state of the Ising model universality class.
As a comparison, we thus also display (Table Figure 3) the best available field theory results obtained from Borel summation of \(d=3\) perturbative series (Guida and Zinn-Justin 1998).